Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188770 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10,12,17,13,18,21,16,14,19,22,24,23,15,11,20);(1,2)(3,7)(4,8)(5,6)(10,15,19,18)(11,21)(12,22)(16,24,20,17);(1,3,5,4)(2,7,6,8)(9,11,13,12,14,21,23,22)(10,16,18,24,19,20,15,17);(1,4,5,3)(2,7,6,8)(9,12,13,21,14,22,23,11)(10,17,18,16,19,24,15,20)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=deh,g^b=gi,d^c=di,g^c=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405004);